Basic principles and description

As a mean to create growing microbial cultures without temporal changes (inher-ent to batch cultivations), the experim(inher-ental concept of continuous cultures was simultaneously developed by the physicist Leó Szilárd who worked with Aaron Novick¹ [62], and by Jacques Monod [58]. The names used by these inventors (’chemostat’ and ’continuous culture’, respectively) are often used as synonyms.

Chemostats fall into the category of stirred bioreactors with continuous operation.

For a modern description see, for example, the textbooks [85], [59], or the review [18].

Since experiments in the systems biology community are typically based on more affordable batch cultivations - such as shake flasks - we will highlight differences between these and chemostat cultivations. The motivation behind this is the question, to which extent may datasets based on different cultivations be compared. An answer to this would be valuable in a field in which integration of multiple datasets is practice.. While an absolute answer is not within the scope of this work, it is hoped that the reader will gain insight to help with decisions in particular cases. Why chemostat cultivations can be expected to result in higher reproducibility of data, will be discussed at the end of this section.

At the macroscopic level, the difference between batch and chemostat cultures is characterised by the ability of the latter to provide acontinuous steady state culture. This implies easier monitoring of many parameters of the culture, such as concentrations, external to the cells. Moreover, one physiological parameter, the specific steady state growth rate of the culture is set independently by the experimenter. Importantly, the chemostat is capable of attaining a steady state (in terms of physiological parameters of the whole culture), a fact corresponding

to the stability of the steady state of the equations discussed here.

A core element of the conceptual framework behind our argumentation is the assumptions that a cell in a cultivation only receives information from its microenvironment. Characterising and, if possible, independently setting this mircoenvoronment is one of the main goals of system biology experiment designs. We will furthermore assume, that changes in the cell’s microenvironment are fully characterised by changes in concentrations - reflecting our focus on processes involving chemical species, rather than quantities such as temperature of radioactive radiation. Consequently, changes in the ’rest of the world’ - such as increasing the glucose influx into the culture - can only influence the cell’s physiology by changing concentrations in its microenvironment by mixing or diffusion. We may refer to this as the assumption of no-fernwirkung² in cell

1Both of them had worked on the Manhattan Project - the Hungarian-German-American physicist-turned-biologist Szilárd having conceived the idea of nuclear chain reaction and contributed to the Einstein letter to Roosevelt - but were, like other involved scientists, disgusted by the way the technology was used to end the War, and campaigned against the use of nuclear weapons afterwards.

2Even though Einstein’s opinion about the ’spooky action-at-a-distance’ in the context of the so-called EPR debate [23] seems to have turned out to be false.

cultures.

While, in our view, the above assumptions describe well the majority of experimental scenarios, they are not consistent with commonly encountered ar-guments based on fluxes into the culturedirectlyinfluencing the cell’s physiology, nor with cell growth in a culture with zero concentration of the limiting nutrient.

The key assumption, necessary for the outlined construction of a con-tinuous steady state culture is that the growth rate of a microorganism is a strictly monotonous function of the provided nutrition concentration³. Under circumstances when this condition is not met - e.g. if the growth rate of the culture has reached its maximum - a chemostat may not attain a stable steady state.

A chemostat may be defined as a bioreactor with certain properties which correspond to well-defined experimental concepts. These involve

• (i) main volume, required to bewell mixed

• (ii) liquid feed

• (iii) limiting nutrient contained in the feed solution

• (iv) effluent, specifying thedilution rate

The following discussion attempts to elucidate these concepts, following an approach somewhat different from other texts encountered by the author ([18], [96], [85], [59]). Striving to increase conceptual clarity, we will explicitly state some necessary assumptions usually made implicitly.

Design principles

The first object from the above list, themain (working) volume, is defined for the purpose of this text as the the liquid phase in the bioreactor, assumed to fully contain the cell culture. Concentrations in the main volume are often called residual concentrations(since often measured in the effluent, as discussed below).

The main volume is supplied by a (i) feed, defined as an incoming flux [mol time⁻¹] of the chosen nutrient composition which is required to be of consant composition resulting in time-independent concentrations and generated influxes of the provided chemical species into the bioreactor.

In addition to the fluid-phase feed, there may be a gaseous feed (controlling, for example, oxygen inflow) usually requiring an exhaust system, to carry away gaseous metabolic products (such as carbon-dioxide).

Since the transformations of the provided chemical species into biomass and products in the bioreactor usually involves mass transfer from or into the gaseous phase, the main volume might vary slightly if the metabolic state of the cells changes. Depending on the particular experimental conditions and the required precision, this can often be neglected.

The main volume is requiredto be(ii) well mixed, for local measurement values of intensive quantities such as concentrations, pH, or temperature

-3In other words, the growth rate increases if nutrition concentration rises, other experimental conditions assumed being constant.

Figure 2.1: Simplified scheme of a chemostat. Of the key concepts, indicated are the mixing, the feed and the effluent. Not indicated but also necessary is the composition of the feed such that a single limiting nutrient is present. Although not requiredper se, gaseous feed and exhaust are necessary in many scenarios to maintain viable conditions.

to describe the whole volume, i.e. the actual microenvironment of each cell.

While this property is often associated with complete spatial homogeneity with regard to these quantities, this is hard to fully achieve in reality. This follows from the fact that cells under nutrient limitation are expected to be sensitive to nutrient concentration differences, while at the same time this concentration will be slightly higher near the feed entering the main volume.

However, it turns out that for many purposes it suffices if each cell experiences the same microenvironment on average. To formulate this requirement in an accurate way, let’s assume that one can follow the trajectory of any small volume of liquid (on the scale of a cell’s microenvironment) within the bioreactor, continuously monitoring all relevant intensive quantities in it.

We will regard a bioreactor to bewell mixed if the following statement holds:

Even if these quantities, monitored in the volume, are not fully constant during its trajectory, their time average should attain the macroscopically measured average value faster than the time scale of those processes which we intend to study under homogeneous conditions⁴.

For example, if the processes to be studied under homogeneous conditions are assumed to have a characteristic timescale of seconds - as it is the case for many signalling events - it suffices to ensure that each cell experiences the same conditions when averaged on a timescale of 100 miliseconds. In practice, this means that the mixing has to prevent the formation of “still areas” which would mix too slow with the rest of the working volume.

However, the mixing also has another purpose. For meaningful measurements to be made, the mixing must be effective enough that a small sample of volume, taken at a pre-defined location in the reactor, reliably gives the average values

4In a more technical - but shorter - formulation, the statistics of each intensive quantity on the ensemble of small fluid volumes is required to beergodicon a time scale below that of the processes of interest.

of the measured quantities in the main volume.

Thus, if the above requirements are fulfilled, we can assume the main volume to appear homogeneous for the purposes of the experiment.

The feed is requiredto supply a(iii) nutrient solution containing one limiting nutrient, Slim (e.g. glucose). The underlying notion is that the con-centration of Slim in a cell’s microenvironment should be the key controlling factor of its physiology while the concentration of other nutrients and products should ideally have negligible effect, at least within the range of the experiment.

Another usual formulation of this notion is that all non-limiting nutrients are to be supplied “in excess” compared to the cell’s needs per unit ofS_lim consumed.

Hence, the property of a certain nutrient being limiting should be regarded as a property of the system rather than that of a single compound.

The experimental definition is associated with the following behaviour of the chemostat: given a steady state of the culture, changing the feed concentration of Slim (hence the influx V_S^{f eed}

lim) while keeping all other parameters constant (including the dilution rate⁵and the feed concentrations of the other nutritients) should eventually lead to a new steady state in which the general state of the cells is the same as in the first stady state. Since this is not a statement easily tackled experimentally, it is usually only required that the biomass composition, the residual concentration ofS_limand the associated biomass yield have attained the same values they had in the original steady state [18].

The(iv) effluent can be thought of as an “overflow” of the liquid phase in the main working volume since it is required to be equal to the latter in composition.

Importantly, the volume per time unit leaving the chemostat through the effluent is required to be a constant fraction of the main volume specifying thedilution rate usually denoted by D [time⁻¹]. The dilution rate should also be held constant during volume changes, which may possibly occur before attaining a steady state. Since gaseous and liquid phases may be interconverted, this condition does not imply that the feed and the effluent fluxes have to be equal.

Note that the above requirement causes the total efflux [mol time⁻¹] of any substance through the effluent to simply to be proportional to its main volume concentration.

Since a chemostat system as defined by (i)-(iv) contains only the quantity D to be set independently and explicitly, the dilution rate is regarded as the main parameter of a chemostat cultivation.

A further independent parameter is the feed concentration of the limiting nu-trientSlimwhich determines the influxV_S^{f eed}

lim for a given dilution rate. However, it follows from the above that this only controls the biomass density, and not the steady state concentrations of external metabolites. This conclusion involves the assumption that biomass density change within a certain range has negligible effects on the cells physiology.

The quantity 1/Dis calledmain residence time since this is the time which infinitesimally small fluid volumes (which may include cells) would spend on

5 Note that if the dilution rate - and hence the steady state growth rate - is to remain constant, the volumetric feed influx must be constant too, soV_S^{f eed}

lim can only be controlled by the feed concentration ofSlim.

Why does D set the specific growth rate? Balance equations and steady state stability.

To understand why a chemostat cultivation attains a steady state, we now discuss the mass-balance equations.

The amount of any chemical species xin the main volume is potentially increased by influx (ifxis provided in the feed), decreased by efflux, and changed by cell consumption or production. The differential equation quantifying the momentarily change of the total amount of x[mol time⁻¹] within the main volume may be written as

d

dtC_x^mainV^main= (2.1)

φ^{f eed}C_x^{f eed} −φ^{ef f} C_x^main −V^mainC_bio^mainv^bio_x +v_x^{gas in}−v^{gas ex}_x where V^maindenotes the size of the main working volume;C_x^main denotes concentration of x in it; φ denotes a volumetric flux [volume time⁻¹]; the superscriptsf eed,ef f, andmainindicate quantities associated with the feed, efflux and main working volume, respectively;C_bio^main[mass volume⁻¹] denotes the biomass density andv_x^bio [mol time⁻¹ biomass⁻¹] denotes the rate at which xis consumed (negative if produced) by the cells. The total influx from and evaporation into the gas phase is summed up in the flux termsv_x^{gas in}andv^{gas ex}_x [mol time⁻¹], respectively.

The temporal change of total biomass,V^mainC_bio^main[biomass time⁻¹], itself may be characterised by an analogous formula, however with a few simplifications:

we assume thatC_bio^{f eed}= 0 (sterile feed⁶), andv_bio^{gas in}=v_bio^{gas ex}= 0 (no biomass influx or loss through the gas phase). Hence we obtain

d

dtC_bio^mainV^main=−φ^{ef f}C_bio^main+V^mainC_bio^mainµ (2.2) whereµ[time⁻¹] denotes the specific growth rate⁷- this could be denoted by−v_bio^bio in Eq. 2.1, albeit biomass is usually measured as mass, instead of mols of cells.

In addition, from the definition of the dilution rate, we haveφ^{ef f} =V^mainD which simplifies Eq. 2.2 to the standard form

d

dtC_bio^main=C_bio^main(µ−D) (2.3) This equation enables us to see that at specific growth rate

µ=D (2.4)

i.e. if cells divide once per residence time on average, the biomass is constant.

6Which is often nontrivial to achieve, since many bacteria can travel upstream with ease.

7A more traditional definition of specific growth rate and its connection to the doubling timetd=log2/µis found in Eq. 2 in [37].

IfC_bio^mainis constant, we can solve Eq. 2.1 for the steady state concentration ofx, denoted by C_x^main, to obtain

C_x^main,sts= 1

φ^{ef f}(φ^{f eed}C_x^{f eed} −V^mainC_bio^mainv_x^bio +v^{gas in}_x −v_x^{gas ex}) (2.5) Hence, steady state concentration of a product (C_x^{f eed}= 0) changes with the total biomass V^mainC_bio^main if its specific production rate is constant. This implies that different feed concentrations of the limiting nutrient will lead to steady states with different product concentrations. Hence the definition of

“limiting nutrient” implicitly employs the assumption that product concentra-tion changes over a certain range are irrelevant for the biological state of the cells.

Stability and uniqueness of the steady state atµ=Dis implied by the requirement of strictly monotonous dependence of the growth rateµon nutrient concentrationsSlim.

The argumentation runs as follows: let us assume thatµ < D. It follows from Eq. 2.3 that this causes a decline in the biomass densityC_bio^main which, via Eq.

2.1, causes the limiting nutrient concentration to rise. Now, the monotonicity condition implies that the specific growth rateµwill increase. Conversely, in case µ > D, an analogous argument predicts the decline of the growth rate.

Hence, the steady state atµ=D is stable and unique.

Note that the above sketched dynamics may not take place if the monotonicity condition does not hold. For exampleD > µ_maximplies that the steady state condition Eq. 2.4 cannot be fulfilled since the growth rate will not increase above µ_maxwith higher nutrient concentration, causing the cells to eventually wash out, resulting in a steady state of limited biological interest. A more exotic example is the case of exceedingly high concentration ofS_limsuch that an increase results in a decrease of growth rate, e.g. due to osmotic stress. In this case, the above theoretical framework predicts the steady stateµ=D(in case it exists) to be unstable, and hence hard to observe in a chemostat⁸.

Stability of a fixed point does not exclude oscillations of the system around this point. Oscillatory behaviour may indeed occur in chemostat cultivations -both as an annoyance and as a feature to be studied. Synchronised oscillation in chemostat cultivations is often exhibited by manyS. cereviciaestrains due to cell cycle synchronisation, and must be addressed if undesirable. Strains of the CEN.PK family, on which data for this work is based, are reported to be less prone to cell cycle synchronisation [18].

The mass-balance equationsabove describe (through the consumption and production ratesv_x^bio) how the biomass influences its environment. However, in order to complete the description, one would need to answer the reverse question: how is the biomass (more specificallyv_x^bio andµ) influenced by its environment?

An important qualitative aspect of the answer was already introduced as the monotonicity assumption for growth rate. This information already allowed to analyse steady states. Moreover, since the assumption is a necessary condition for the prediction of the - experimentally observed - stability of steady states, it can be regarded as an experimental fact within the range of observation of

8Regulated dilution rate would presumably make observation of such steady states possible.

“limiting nutrient”: the assumption that as long as the concentrations ofS_lim is the same in two steady states, cells in these cultures are (practically) identical, independently from the other concentrations.

Ideally, a complete, quantitative answer to the above question would allow to predict the dynamic behaviour of the chemostat culture from any given initial state. However, this would require complete knowledge of the relevant biology. Hence there are only partial answers available, typically in the form of approximate, phenomenological formulas.

In many cases the assumption is employed that the growth rate µgenerally (not restricted to constant growth rates) only depends on the concentration of

the limiting nutrient. Often the Monod-equation⁹ µ=µmax

Slim

Slim+K (2.6)

is used to quantify this dependence, whereµmaxand Kare phenomenological constants depending on the experimental conditions. Note that this equation ful-fils the monotonicity requirement for the growth rate, and exhibits an asymptotic maximal growth rate,µmax. Naturally, such a phenomenological description has a limited range of validity, for example the prediction to associate any near-zero substrate concentrations with non-zero growth rates is of no direct biological meaning.

Equation 2.6 also allows to estimate the dependence of the steady state residual concentration of the limiting nutrient, which we denote withS_lim^stst. The resulting expression is

S_lim^stst= K

µmax/D−1 (2.7)

Again, predictions for S_lim^stst withD/µmax near zero or near one are to be treated with caution. Nevertheless, the general trend is intuitive: the residual steady state concentrationS_lim^stst steeply rises as the dilution rate comes close to the maximal growth rate of the microorganism under the given conditions. This was indeed reported in [96] (s. Fig. 6).

The above drawn pictureis of course simplified: in real life, experimental (and financial) limitations are present. Hence, in practice it is usual to monitor only biomass density, as well as concentrations and external fluxes of a few selected chemical species. A practical definition of steady state is to regard at least five main residence times as necessary to reach steady state, which is assumed to have been reached if macroscopically monitored quantities change less than 2 % whithin the next main residence time [18].

Comparing cell cultures from chemostat and from batch cultivations should be undertaken with great care, since the two methods present different microen-vironments to the cell. In a chemostat, cells experience a continuous growth control by limitation of a single nutrient, while in typical batch cultures no initial nutrient limitation is present, usually resulting in higher growth rates during the exponential phase of the culture. There may be scenarios during the lag phase of a batch culture when nutrient concentration has diminished to be comparable

9In order to increase readability, concentration of the limiting nutrient will be denoted simply bySlim(instead ofC_S^main

lim or [Slim]) in all formulas.

Im Dokument On the regulation of central carbon metabolism in S. cerevisiae (Seite 14-21)